Continuity at a Point and on an Open Interval <ul><li>Informal definition of continuity: To say a function f is continuous at x = c means there is no interruption at x=c; no holes, no jumps, no gaps. </li></ul>
Discontinuities come in two types – removable and nonremovable. . . .Removable if f can be made continuous by appropriately defining or redefining f(c) , that is, by filling the hole. . . .Nonremovable if there is an asymptote or gap, not just a hole. If a function f is defined on an open interval I (except possibly at x=c) and f is not continuous at c, then f has a discontinuity.
Ex 1, p. 71 Continuity of a Function Discuss continuity: Domain: ? Is it continuous at every x-value in its domain? Is there any way to fill the discontinuity?
Domain: ? Is it continuous at every x-value in its domain? Is there any way to fill the discontinuity?
Domain: ? Is it continuous at every x-value in its domain? Notice as x ->0, the limit is 1, so they connect
Domain: ? Is it continuous at every x-value in its domain?
One-Sided Limits and Continuity on a closed interval
When limit from left is not equal to limit from right, the (two-sided) limit does not exist
Idea of one-sided limit extends definition of continuity to closed intervals. It is continuous on closed intervals if it is continuous in the interior and exhibits one-sided continuity at the endpoints.
Ex 4 p. 73 Continuity on a Closed Interval Discuss continuity of Domain is [-1, 1] (closed) Continuous on interior Is it continuous?
Ex 6, p.75 Applying Properties of Continuity Are these functions continuous? Why or why not?